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Properties

Label 2-252-7.2-c3-0-0
Degree $2$
Conductor $252$
Sign $-0.717 + 0.696i$
Analytic cond. $14.8684$
Root an. cond. $3.85596$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−9.58 + 16.5i)5-s + (−6.16 + 17.4i)7-s + (−13.2 − 23.0i)11-s + 10.3·13-s + (−50.6 − 87.7i)17-s + (46.8 − 81.1i)19-s + (−4.28 + 7.42i)23-s + (−121. − 209. i)25-s − 52.3·29-s + (27.7 + 47.9i)31-s + (−230. − 269. i)35-s + (−214. + 371. i)37-s + 137.·41-s − 172·43-s + (−24.6 + 42.6i)47-s + ⋯
L(s)  = 1  + (−0.857 + 1.48i)5-s + (−0.332 + 0.942i)7-s + (−0.364 − 0.630i)11-s + 0.220·13-s + (−0.722 − 1.25i)17-s + (0.566 − 0.980i)19-s + (−0.0388 + 0.0673i)23-s + (−0.969 − 1.67i)25-s − 0.335·29-s + (0.160 + 0.278i)31-s + (−1.11 − 1.30i)35-s + (−0.951 + 1.64i)37-s + 0.521·41-s − 0.609·43-s + (−0.0764 + 0.132i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.717 + 0.696i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.717 + 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.717 + 0.696i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ -0.717 + 0.696i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.07000006672\)
\(L(\frac12)\) \(\approx\) \(0.07000006672\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (6.16 - 17.4i)T \)
good5 \( 1 + (9.58 - 16.5i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (13.2 + 23.0i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 10.3T + 2.19e3T^{2} \)
17 \( 1 + (50.6 + 87.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-46.8 + 81.1i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (4.28 - 7.42i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 52.3T + 2.43e4T^{2} \)
31 \( 1 + (-27.7 - 47.9i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (214. - 371. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 137.T + 6.89e4T^{2} \)
43 \( 1 + 172T + 7.95e4T^{2} \)
47 \( 1 + (24.6 - 42.6i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (237. + 410. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-98.5 - 170. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-200. + 347. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (62.7 + 108. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 788.T + 3.57e5T^{2} \)
73 \( 1 + (302. + 523. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (391. - 678. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 339.T + 5.71e5T^{2} \)
89 \( 1 + (-255. + 443. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 672.T + 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.73108650449523871136598314778, −11.46159464908533445959822954122, −10.45210199212828827550164133742, −9.303207940475065237708681854795, −8.244732836903609162887837826654, −7.12835938537618783374176348263, −6.39474123395813983443199172590, −5.01703318759105533670282244546, −3.33630791522076643075068287972, −2.65320498276553849061212475572, 0.02804105749147295303977499782, 1.48054364312574821224525611490, 3.78418052550444808790840383036, 4.45483433597741805502673295044, 5.73712375630595232853011152918, 7.24589116711710948892923864963, 8.056155979452303324367411679704, 8.950395437519570362260014707813, 10.04362657831978322865360004870, 11.01641846903463625564756211880

Graph of the $Z$-function along the critical line